Mathematics and Visualization – Rajni Ranjan

Nature around us is at times too beautiful to view and too complex to understand simultaneously. But the explanation can be simplified by the use of mathematics. For centuries, math and the visual arts have been intricately connected, although most of the times, it is not simple to understand the principle behind that. In fact, it is the amount of overlap between the two which our brain finds so interesting. Often, a colouring algorithm can produce automatic art that may be as surprising or aesthetically pleasing as that produced by a human hand. There have been attempts to resolve the apparent difference between these two seemingly contrasting fields, which this article aims to explore.

Perhaps, one of the most important connections between art and mathematics is the enormous use of the number Φ=(√5+1)/2, better known as the ‘Golden Ratio’. It is very useful in the depiction of human figures because of the expression of this ratio in dimensions of body parts. Also, the incorporation of this number in drawings, such as the golden spiral, gives them a divine appeal, hence it is also sometimes known as God’s own number.

Geometric designs were extremely useful tools to express art in medieval and ancient times. The repetitive nature of these complex geometric designs suggests the infinite power of God.

The Alhambra tilings, constructed by Nasrid, the last of the Mughal emperor in Spain, are periodic and repeat themselves after certain distances. The mathematics of kaleidoscopes, based on symmetry, is so appealing that it has attracted many a scientists to work in that field. Roger Penrose, an Oxford professor invented the most famous set of tiles, named after him, which can be grouped into `vertex stars’ (tiles sharing the same vertex). These tilings can cover the entire space aperiodically which is a phenomenon called quasiperiodicity.

Often, in such a tiling, the ratio of thick to thin rhombi is φ , again the `Golden Ratio’. Another form of symmetry can be observed in the regular polyhedra, which are three dimensional solids in which all faces are identical. Their symmetries can be represented by a group structure. Hence a concept as simple as a self contained system of symmetries, opens the gateway to abstract algebra.

Fractals which are yet another class of visual representation depict high levels of symmetry, more specifically, in self-similarity. It means, no matter how closely it is zoomed in on, the pattern is highly complicated and crinkly and looks similar to the whole fractal itself.

Mathematical formulae, better depicted by art are comparatively easy to understand. Topology is such a branch of mathematics which is almost entirely based on visualisation. The Möbius Strip, which is a one-sided surface, has the boundary of a closed curve. It means that if we start painting the strip from any location, the entire strip gets painted without having to change the sides. Klein’s bottle, again a one-sided surface can be visualized by inverting a piece of rubber tube and by letting it pass through itself so that the outside and inside meet. In addition, coloured versions of the Mandelbrot set and Julia sets, each generated by a set of recursive equations in the complex domain are equally striking. For example, the Glowing Gasket, which is a two dimensional pattern is generated by very simple complex matrices.

Projective geometry, a very useful tool for artists for understanding perspective, helps in incorporating depth and illusion in their artworks. In this approach, unlike Euclidean geometry where only two points define a line, it is assumed that each straight line has at least three points with one focused at infinity. Hence, no two lines are parallel. Donald Coxeter and M. C. Eschers who work on non Euclidean geometry are a marvel in the field and have also influenced art to a great extent.

Several other sciences also use this connection. Anamorphis is a technique in which an image is mathematically distorted so that it makes sense only when it is viewed along certain angle. It is a good demonstration of the mathematics of transformations. Origami, the Japanese craft of creating exquisite three dimensional shapes solely by folding paper utilizes complex geometrical patterns of creases made up of triangles and squares, many of which are congruent.

The incorporation of computational techniques in mathematical visualisation has helped in the exponential growth of the visual media. For instance, the animated movies we see are based upon algebraic equations over a complex domain. Also, for depicting 3-dimensional rotations, special type of numbers called quaternions, are used. The 360° views used in computer games, and for example, for displaying tourist sites like Taj Mahal, utilise the principles of projective transformations. In this method, three points are considered in the overlapping portion of both the images and a projective transformation is mapped from and onto these points. This helps in effective merging of the two photographs and hence creating 360° view.

Yet, computers are still to understand the mechanism of the human brain which transforms the 2-dimensional images built on our retina to visualise a 3-dimensional world. Hence our brain is able to solve, in although an unexplainable way, a very difficult problem that most of the computers still have trouble with.

Mathematics has given a new dimension to the definition of visualisation which will undoubtedly be of tremendous importance in the long run. The boundaries of visual art and mathematics have now grown fainter and the intricate connection between them is clearly visible today.